%I #41 Jan 28 2024 23:27:05

%S 0,0,1,0,1,0,0,1,1,-3,0,1,2,0,-8,0,1,3,3,-2,-15,0,1,4,6,4,-5,-24,0,1,

%T 5,9,10,5,-9,-35,0,1,6,12,16,15,6,-14,-48,0,1,7,15,22,25,21,7,-20,-63,

%U 0,1,8,18,28,35,36,28,8,-27,-80,0,1,9,21,34,45,51,49,36,9,-35,-99,0,1,10,24,40,55,66

%N Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.

%C Note that the formula gives several kinds of numbers, for example:

%C Row 0 gives 0 together with A258837.

%C Row 1 gives 0 together with A080956.

%C Row 2 gives A001477, the nonnegative numbers.

%C For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Polygonal numbers</a>.

%H The OEIS, <a href="http://oeis.org/wiki/Polygonal_numbers">Polygonal numbers</a>.

%H University of Surrey, Dept. of Mathematics, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Figurate/figurate.html">Polygonal Numbers - or Numbers as Shapes</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FigurateNumber.html">Figurate Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%F T(n,k) = A139600(n-2,k) if n >= 2.

%F T(n,k) = A139601(n-3,k) if n >= 3.

%e Array begins:

%e ------------------------------------------------------------------------

%e n\k Numbers Seq. No. 0 1 2 3 4 5 6 7 8

%e ------------------------------------------------------------------------

%e 0 ............ (A258837): 0, 1, 0, -3, -8, -15, -24, -35, -48, ...

%e 1 ............ (A080956): 0, 1, 1, 0, -2, -5, -9, -14, -20, ...

%e 2 Nonnegatives A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...

%e 3 Triangulars A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, ...

%e 4 Squares A000290: 0, 1, 4, 9, 16, 25, 36, 49, 64, ...

%e 5 Pentagonals A000326: 0, 1, 5, 12, 22, 35, 51, 70, 92, ...

%e 6 Hexagonals A000384: 0, 1, 6, 15, 28, 45, 66, 91, 120, ...

%e 7 Heptagonals A000566: 0, 1, 7, 18, 34, 55, 81, 112, 148, ...

%e 8 Octagonals A000567: 0, 1, 8, 21, 40, 65, 96, 133, 176, ...

%e 9 9-gonals A001106: 0, 1, 9, 24, 46, 75, 111, 154, 204, ...

%e 10 10-gonals A001107: 0, 1, 10, 27, 52, 85, 126, 175, 232, ...

%e 11 11-gonals A051682: 0, 1, 11, 30, 58, 95, 141, 196, 260, ...

%e 12 12-gonals A051624: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...

%e 13 13-gonals A051865: 0, 1, 13, 36, 70, 115, 171, 238, 316, ...

%e 14 14-gonals A051866: 0, 1, 14, 39, 76, 125, 186, 259, 344, ...

%e 15 15-gonals A051867: 0, 1, 15, 42, 82, 135, 201, 280, 372, ...

%e ...

%Y Column 0 gives A000004.

%Y Column 1 gives A000012.

%Y Column 2 gives A001477, which coincides with the row numbers.

%Y Main diagonal gives A060354.

%Y Row 0 gives 0 together with A258837.

%Y Row 1 gives 0 together with A080956.

%Y Row 2 gives A001477, the same as column 2.

%Y For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).

%Y Cf. A139600, A139601.

%Y Cf. A303301 (similar table but with generalized polygonal numbers).

%K sign,tabl,easy

%O 0,10

%A _Omar E. Pol_, Aug 09 2018